Commutative quasigroups and semifields from planar functions

Abstract

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most $|N_{\mu }:{(N_{\mu })}^2{N}_{\lambda }|$ classes with respect to the strong isotopy, where $N_{\mu }$ and $N_{\lambda }$ are the middle and the left nucleus of Q. Loops Q₁ and Q₂ are said to be strongly isotopic if there exists an isotopism $Q_1\to Q_2$ of the form $(\alpha ,\alpha ,\gamma )$.

KEYWORDS:

• Commutative loop
• commutative semifield
• Dembowski-Ostrom polynomial
• nucleus
• planar function
• semifield plane

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 12K10
• Secondary: 20N05